However, even on Earth there would be no crater. Tests from lander prototypes did not produce any such crater, and neither do routine landings by military VTOL (Vertical Takeoff and Landing) aircraft. A small amount of disturbance maybe, but certainly nothing like a crater. Bear in mind that Harrier jets are far more powerful than the lunar lander and they don't go around making craters.

Furthermore, dispersal of dust on the ground is caused not by rocket exhaust, but by the displacement of air. There is no air on the moon, therefore no significant dust movement.

There is however a small effect on the surface. The photo below shows a close-up of the ground directly under the rocket nozzle. You can actually see where the exhaust has caused discolouration and an outward sweeping effect. These effects can by duplicated on Earth using simple rocket equipment.